Brill-Noether theory and Green's conjecture for general curves on simple abelian surfaces

TL;DR

This paper extends Brill-Noether theory and Green's conjecture to curves on simple abelian surfaces, using vector bundle techniques adapted from K3 surfaces, and provides new results on gonality and Brill-Noether loci.

Contribution

It introduces a method to compute gonality and Brill-Noether loci dimensions for curves on abelian surfaces, confirming Green's conjecture in this context.

Findings

Gonality of curves on abelian surfaces determined

Dimensions of Brill-Noether loci computed

Green's conjecture verified for these curves

Abstract

In this paper we compute the gonality and the dimension of the Brill-Noether loci $W^1_d(C)$ for curves in a non primitive linear system of a simple abelian surface, adapting vector bundles techniques \`a la Lazarsfeld originally introduced with $K3$ surfaces. As a corollary, we obtain general Green's conjecture for curves on abelian surfaces.